Dualities in quantum integrable many-body systems and integrable probabilities. Part I

In this study we map the dualities observed in framework of integrable probabilities into familiar a realm many-body systems. The between pairs stochastic processes involve one representative from Macdonald-Schur family, while second is higher spin six-vertex model TASEP family. We argue that these are counterparts and generalizations quantum-quantum (QQ) One system QQ dual pair belongs to family inhomogeneous XXZ chains, Calogero-Moser-Ruijsenaars-Schneider (CM-RS) wave functions Hamiltonian CM-RS known be related solutions (q)KZ equations at chain side. When function gets substituted by measure, bilinear functions, similar correspondence holds true. As an example, have elaborated some details new duality discrete-time multispecies on circle quantum Goldfish RS present precise 5-vertex trigonometric rational models respectively, where local jump rates get identified as coordinates model. Some comments concerning relation with SUSY gauge surface operators included made.